Estimating or Calculating: Deciding What a Principal Can Trust
The third grade runs a book fair. On Monday, 604 tickets are printed and 287 are sold. On Tuesday, 198 more tickets are printed and 356 are sold. The principal asks, about how many tickets are left. Aria rounds each number to the nearest hundred, computes 600 − 300 + 200 − 400, and reports about 100 left. Ben works through the days step by step and gets an exact count. The principal then says that she needs to know whether the fair can still serve 175 more students. Aria's estimate and Ben's exact answer happen to lead to the same yes-or-no answer, but the teacher points out that this agreement was luck rather than method, and asks the class to find a rounding pattern that would have produced an estimate larger than the true number of tickets remaining. Students must also identify which rounding decisions in Aria's work pushed her result below Ben's.
1. Carry out both computations and explain the gap between them by naming each rounding decision and its effect.
Read the worked answer
Ben: 604 − 287 = 317 tickets left after Monday; 317 + 198 = 515 after Tuesday's printing; 515 − 356 = 159 left.
Aria: 600 − 300 + 200 − 400 = 100.
Every one of the four numbers was moved, and each move shifts the result in a direction: • 604 rounded down to 600 lowers the estimate by 4. • 287 rounded up to 300 lowers it by 13. • 198 rounded up to 200 raises it by 2. • 356 rounded up to 400 lowers it by 44.
Net change: −4 − 13 + 2 − 44 = −59, and 159 − 59 = 100, which matches Aria's report exactly.
2. Which number should the principal rely on to decide whether 175 more students can be served, and how could a different rounding choice have produced a misleading yes?
Read the worked answer
Rely on Ben's 159. Since 175 > 159, there are not enough tickets for 175 more students.
Aria's 100 happens to give the same no, but that is not a guarantee; her rounding moved the numbers in mixed directions and could just as easily have landed above the truth.
A rounding pattern that overstates the supply is the dangerous one. Round the tickets printed up and the tickets sold down: 610 − 280 + 200 − 350 = 180. That estimate is larger than the true 159. If the principal needed only 165 tickets, this inflated estimate would answer yes while the truth answers no. For a do-we-have-enough question, an estimate that is too high is the kind that causes a real shortage.
3. Design an estimation plan that is guaranteed never to exceed the true number of tickets left, and show it on these numbers.
Read the worked answer
To force a safe underestimate, shrink what is added and enlarge what is subtracted: round printed and added amounts down, and round sold and subtracted amounts up.
Worked out: 600 − 300 + 190 − 360 = 130. Every step went in the safe direction, so 130 ≤ 159 and the estimate cannot be larger than reality.
How to use it: if even 130 tickets would cover the need, the answer is certainly yes; if 130 is short, the class cannot yet conclude no and must compute exactly. The estimate is a screening tool, and the exact count (159) settles the question.